Magnetohydrodynamic drift equations: from Langmuir circulations to magnetohydrodynamic dynamo?

TY - JOUR

T1 - Magnetohydrodynamic drift equations

T2 - from Langmuir circulations to magnetohydrodynamic dynamo?

AU - Vladimirov, Vladimir

PY - 2012/5

Y1 - 2012/5

N2 - We derive the closed system of averaged magnetohydrodynamic (MHD) equations for general oscillating flows. The used small parameter of our asymptotic theory is the dimensionless inverse frequency, and the leading term for a velocity field is chosen to be purely oscillating. The employed mathematical approach combines the two timing method and the notion of a distinguished limit. The properties of commutators are used to simplify calculations. The derived averaged equations are similar to the original MHD equations, but surprisingly (instead of the commonly expected Reynolds stresses) a drift velocity plays a part of an additional advection velocity. In the special case of a vanishing magnetic field $h\equiv 0$, the averaged equations produce the Craik–Leibovich equations for Langmuir circulations (which can be called ‘vortex dynamo’). We suggest that, since the mathematical structure of the full averaged equations for $h\neq 0$ is similar to those for $h =0$, these full equations could lead to a possible mechanism of MHD dynamo, such as the generation of the magnetic field ofthe Earth.

AB - We derive the closed system of averaged magnetohydrodynamic (MHD) equations for general oscillating flows. The used small parameter of our asymptotic theory is the dimensionless inverse frequency, and the leading term for a velocity field is chosen to be purely oscillating. The employed mathematical approach combines the two timing method and the notion of a distinguished limit. The properties of commutators are used to simplify calculations. The derived averaged equations are similar to the original MHD equations, but surprisingly (instead of the commonly expected Reynolds stresses) a drift velocity plays a part of an additional advection velocity. In the special case of a vanishing magnetic field $h\equiv 0$, the averaged equations produce the Craik–Leibovich equations for Langmuir circulations (which can be called ‘vortex dynamo’). We suggest that, since the mathematical structure of the full averaged equations for $h\neq 0$ is similar to those for $h =0$, these full equations could lead to a possible mechanism of MHD dynamo, such as the generation of the magnetic field ofthe Earth.

KW - dynamo theory, general fluid mechanics, magnetohydrodynamics

UR - http://www.scopus.com/inward/record.url?scp=84859977826&partnerID=8YFLogxK

U2 - 10.1017/jfm.2012.40

DO - 10.1017/jfm.2012.40

M3 - Article

SN - 0022-1120

VL - 698

SP - 51

EP - 61

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

IS - n/a

ER -